Yup, that's correct.
Maybe it depends on the field and the country, but I know some industrial PhD position pays a better salary than the one employed full time by the university. Not sure if it is considered to be "much", but at least around 20% more.
The Industrial PhD pays the same here, so that’s mostly a position you would take if you specifically want to collaborate with a company on some research project, not something you do because of income considerations.